BSTJ 50: 3. March 1971: The Energy and General Translation Force of Cylindrical Magnetic Domains. (Thiele, A.A.; Bobeck, A.H.; Della Torre, E. ; Gianola, U.F.)

6.2 MB·English

THE BELL SYSTEM
TECHNICAL JOURNAL

Tae Wark Near

The Energy and General Translation
Force of Cylindrical Magnetie Domains

By A. A.TIIIELE, A. HRORFCK, F DBLLA TORRE,’ and
UL RGTANORA

Stammipt vind Over F190)

1 thie paper we compute the change is the enevay of w foray
vwoxnetioed wmiezal plaeiet roduren by the intradvetinn of « cutie
rica! domain. Difreatinn of the energy expression sible the
leanvational fores produced by gendirate im plate thickness, mateint
conpsiton, temperature, Phe fore expressions provi @ mea for

hese narameters om the margins af
f the gradient produced forces to
fied Soe tes pons etsy spc TR sing
raul ave prevented in bath graphical ad tabular form

‘Magnetron svt, donna
in unico elatle:s have ecoeat'y rveived qouniderable ation? =
‘The thvary of he sate wabil-ye: ove Sma? ad is eppheation
Ty mere! des doves! ave ow Sarno in peevietspepen.
‘This paper is coxserned willy Je lierslaiona Saree setirg bn the
leaned thes eel an evi pean

nore and lagi devices emstoying
712 rue mau svsnine wen sioa, goes, waKON ToT

1a most device spplientions,aylindrieal domains a propagated by
raciens in the applied fel Cyreslomaiza may, bowser, be
propagate iy praients io any ofthe indeper de paramere whieh
{termine she iota dovinneasegy, Taste parmnrene ze: che applet
fio Hy the pnts erro, By fa saturation magne: zation, M,
tod the wa eretey deusky oy. The dorian sada, vy, is r08 4m
Invlegendeat pausntes for Zmravae in equim but i ewes
‘ones the alter pacasietace are epee) ‘Te gradient ix ae amd Wy
iney be procugel by snmestin gsden:s, sin qtadiente or “me
Dovaane gradin:s, Compan ot troknese gradivesa ey Be ued to
provide foroee which tre Condon: af postion only, whi temperate
tr stein eraens way Ie ms fo poovide ine variable fees.

The tranlational foro in ubtsined by diorentation of the ‘otal
domain energy expression with respeet to pasion, under the wssump
‘on thatthe eredienls in Une donia parameters whieh proce the
‘raneadional force ace sulleeatly amall thet the domain remeing
fdzeulae and ete; wonstqueny the enengy expression remains vali
Sinee this method of eomputing the force i independent of Ue te
tall stoess pecan whic prslue the domain mieten, mo estimate
bf the shape distorting lotlenes 6” th varies parameter is obuaims!
que ting the sim of the raneletiunel fonees un she diag force pices
“he soneral velily squason, ard exsyparing che magsiudes of the
~arious fore: vied tier efeot undies apertion,

‘The energy change procueed by the intzaduetion ofa single laced
circular 180° domain into an in’wite plaia of uwiax‘al raagpetic
tern hie, ie otherwise uniformly mags! along, the average
pave normal (the? axis) fs wow esloalatod. Soch » demain eonfigurs-
tom is how in Fig I. The acungptions and notation of Kets 3,4 and
[are naintined exepe tl we dinain poxemetars h 2 My ey 2d
ry rte allowed Uo be Tunetions of position on the ple. Ta pusticul,
the following is eeeame in the acodel: tho wall fas odie ith,
the wall is everywhow alle othe 2 axis, end the wall enor
Aenity is inepelent 0 wall aventabon or survasure. The values uf
all parameters ave asetines 29 vazy sufieiertly slowly tha: chey say
bo tepecented fy thelr eeavaragel voluee tthe ester of the ere
omen, Adclienally, che azplial field ie eeweaertad by Ss © eam
ponent, if since cine the asscmnplivis seated sore only aie goat
Donent into-nels wil tl sarin,
nana Maeset4e wna ma

sense of

"The toed sheage in the onecey ofthe eystam Gus to the 9
As cylindnea! demain

Tbe Deate, 42Attath — Seb ear reOnse. a)
4 this expres the fret term, the wll eneney, i the product of
the wall energy density and the wal ance; the eecond tern, the applied
field inteacion energy, ie the producl af Uae magnetization edange
24,, the typed fel and the damvain volume: aval the thi tern,
tne inna seagueostate energy, iv the negative of Be inrgral of

Ihe eenerlne si naactotae forge of Le. The inennel nate
netotate <nrgy funtion, TE2y'hy is therelnee dened as

onsn = f°" Pate @

ware Flak is ube farce Cunetion defined hy equntons $8 one 198 of
et 8 The loner init f the inegral £2) lun so Ua When Ha
plate is aniferuly tuggnetied y= 0, che dain energy expres
(2) ie ero, Varous closed form and power svie roprsentatios of
a/b) sxe given in the appendix of this paper, This fuaetion, which
is plowed in Fig. 2 us a faretin of the diameter-tnhilonos rato,
{/h, ad is tshutaed in Table T, has Gre wernt forme
AOE 10 to

2G) +8G)]. fxn om
“Fe nso rcn of she appt

sisuneed in coro detail in Seetion IV
"Ta domi energy per smit mall Jeo ie

le}. Bes coo

jr of Ue £ fanetion was

Eqfod — CeMEN — FQN) o

‘bere 1 = strife the ekursetaiaio macaral langt, where the
sppled fk! hus heen eliminated using Use eoulibviar, condition? *

and where
leeNptest, Maes FOI6 NAN ne

i ig
iB Ye
‘Bo te
ee

eSbe 285)

BEESES:
TS wmaniz eretesemucemicat Joma, MANGE nL

“The inet J hes the axympcocie forse

Q-4-20)+ +20] elt bo

ai) sai) -

The function J(d/l, the normalized total magnetostatic energy per

‘ é
4@), fxn 6)

nit wa gt ie plteed in Hig. 3 logelir with che fore fonction
F(d/h) nt the stbility fanetone S472) and S(a/A) which fave
the aymptotie forme

wlih ber oo

aap 1fay a
-2G) BG). i (

[Namerisal vale ofall how notions are given is ‘be I.
ines fom the Miguie we the asymptetie forts of the faretions
(0), 18) and (9), JC/K) Tes rghiy micway (we respect to dara
ties) betmeen (4/8) and Ss @/h} and sacs the wort for domain
sabiicy ie S,1d/A) > Uh > SiC0/R), chen platelet of arbiteary
‘hikaewe may alway Se bee wh shal the ncvetion ofa comet
Fig, She marwrnvace nan gen rit well enh finn J the

Lacoste fas ftction, Fae The neorettaie seatey archon, Rnd By
Sdom of Usuiewiesteabisthes ri

reduce sicher ¢ positive, oF noxntive, oF wero eng in the eotal
trevey. The eres ‘enlian, PUd/2, one aloe i he Ste so Tat
the bce felesyiiding these eonetions maybe coterie, he vaes
nf Um eunelore ce! its fells uf sero wnt’ decermized owing Pig
So Table L egotber wes i) ane I5r are awl hegre wil Cae
fitehied prvvieus. Tn the ease of a domain having tas predered
tiotenionse? 1h = O50 ane d/h — 200 eornespandine toe £>
ce eld oP Hyfe24, = 0270), chen J120001 = 12492. Since J
hey equal 0 {he oral energy is eeanly av, Unde che prefered
oltione an! ina plate iy which Away — 160 Clause and ¢— 10
Inierosy the absolute value né the wal euergy and the total mag
netastaie cncrey change are cach upprusimatsly 02 tines the est

ene af Ue eet [Nei the ere in Er’ aot be ee
Fed as set energy and mopnetadtetio eontributins ease the
uulign eondion Was used eliminate the applied eld]

"The can

tonal fore i> given by

Fe vB,
118 mer gens enero mearvrew soma, ase

(8)
(Bo

Mare desea

Ve 0)

Ta hia expression H, A, and ne a ein independ
‘Fviables (neti of positing on the platelet) andy fa dependent
finction of thaw rarable determived by the equilib condition (
Since the fundareatal equilib enaition i (350 eos ~ Oy
the lst term in enue (0) ay be dropped. Hvaluating the remain
ing term ring eauaions (1) and (2) yields

= —Deeree | 230 oerh — Gea
f AarhOeM F/I A Mae AI VIE
[erbh — ADEM IR DITA,
= ersve, an

Blinineting the applied fila vain the equilibrium eondktion (5) and
reeraning yelde

Baw QWQeaF| Uh — RAPOIVID EUR — We NTH
A OLARTVANLI, ~ AYNTenfe.y

n=l) ~~ lf) ae
80 +f 20) mb. Ber aw
oil) am fen, «sg

ni oi ws
Bed i +bt@Tolt. 5

4 ea, sy
+h Q fas 059
enim taieiae asain ne

Ligure 4 saows pls of Fy, Py mul P= Pe Znetons of the wort
fuged domain dhameess. Numer sl value af tari duetieey ane i
hea in Tate T. From che Sigize and the aoyrapti for of (Th

{B), 18 wu 114s, is ses that the tunotons are sine se he
Soli aunt Ph are gaeater thea Slats lm Hse paetien a2
Teel eine SyL4/P) 2 13h Sei) se me Ma fe
fy ea ble nwo lve iekess graces ao aus ou goadies
onerbations tthe fetes are ste Eseetice 2 tho gradient while
he deld gradient aud wa |v sy sadist eontrondione wo Ue ae

fare in ree dren uppity tte palit
"The value vleall wae af the tems ir. cowaton (22) esl bat
saecetroals 1 Head [no defined the olz9%2 ar lipo nn

ut Beles eeeectivey Un Piswnsoni a grace fe ze soe eb
sree ths stab ey dite re"naan rocahly ler Ref the Freese
thuecc on a domain fae whack «ik — 2, be — 1 VR = ly ~ B16,
(yet trons an Ag’, ~ 100 Gs x apsraaimately B10
thas WH nueh theokive derve meearenens volt pesly be
azned oul, € nore levine sper foro ates ines the
fealsoeing of (1° yoaiontsagaicst greens fu he oer pw ies
Avthe prone tia “toe mesure nena have beer. sompszed. ew

ther Ihe srestions of Eze api 4M gradient eyes be sll energy
tion fenee, ge Dinos sr fone hae heen vee, Te
gu of che 1 pendent. toes fs were 43 eveseay dwhe ypertinn,

TE ta He
FO) turns. sven TRNAS, SRE en

‘The sien of oy nant Fores we choral csog «tuse-v.une seat
sna ample of Sta, oeTbyal ely. Aetordicg to che dca of TC.
Tove st ream vampavetie she yell aneage camoexature eeeicient
fn epproximesely °1.49¢/C" axe the mepazizction = only ely
theming “clon of envetnline, The on etalien Troe Lie doe
nates in thie waters], ard av ghartwed thet the domairs mere
fH rn Igoe ye merce cams ions oe aad. The
sien of Ure My goulont noe see cent ine a leer
roves in a saints of Gag, BrsOy, earn. Ts this maceriel 30,
Fnemaage ae Se rare ennai elle Te TL energy
soprosimacly seman The 3 3

secrvad Ua ube Comains moved or

ion frre tv nina ed
“easné eglone am neict.

The fre suc ane uel pe
SF nes in we lesa a ls ov tang
thelbnk acct slate! when rears:
Tien eth yy ve expla
spervat J say “oe mmr eae Te ign Ske
‘our the Lada ers ore won DPPH ext
The tre toe toirg rw comean cojjang er orm
colons in equten 88 0f Ret 8 i dr by
p> faa [Eas Zon] a8

sei Hic the wall oon society ae te wall on
itty pateg” ant obama ne 03 ane te

dean fowee (IG) sive the vel waqsati on at ocbervise Seip
‘vanagating dona, > encer boi tae-weee stn i thin caste,
{Vi ees tae Ud lle gration ie ie Te same ie

to ann areseilise Ty way errvenal tp acta Kel Lege
faseueTarm sa hal ee magnon teas expres seme a

To ruin nn saunter aire eve ve a ed Lp Che
ARorain Phamien, GS — BAGH, Thee Lene sasunsedne‘e veoty
‘sjuatior.

ah

ag ae
nett eo ae ve
a-A[E al] am

- fb afs)], are